NUMERICAL SOLUTION OF LONGITUDINAL AND TORSIONAL OSCILLATIONS OF A CIRCULAR CYLINDER WITH SUCTION IN A COUPLE STRESS FLUID

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1 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com NUMERICAL OLUTION OF LONGITUDINAL AND TORIONAL OCILLATION OF A CIRCULAR CYLINDER WITH UCTION IN A COUPLE TRE FLUID J. V. Ramana Muy, G. Nagaaju and P. Muu Depamen of Maemac, Naonal Inue of Tecnology, Waangal, A.P., Inda E-Mal: jjoyula@yaoo.co.n ABTRACT Te flo of a couple e flud geneaed by pefomng longudnal and oonal ocllaon of a poou ccula cylnde ubjeced o conan ucon a e uface of e cylnde uded. A fne dffeence meod popoed o analyze e elocy componen, n an nfne expane of an ncompeble couple e flud unde anng couple ee of ype A condon o upe adeence condon of ype B on e bounday. Te effec of couple e paamee, Reynold numbe and e ao of couple e coe paamee on anee and axal elocy componen ae uded. Te dag foce acng on e all of e cylnde deed and effec of couple e paamee on dag ae on gapcally. Keyod: couple e flud, longudnal and oonal ocllaon, dag, ucon, njecon.. INTRODUCTION In numeou ecnologcal applcaon, e flud n ue do no obey e commonly aumed lnea elaonp beeen e e and e ae of an a a pon and e accuae flo beao canno be pedced by e clacal Neonan eoy. uc flud ae ecognzed a non-neonan flud. In pacula, e nee n non-neonan flud a gon condeably, due lagely o e demand of uc dee aea a boeology, geopyc and cemcal and peoleum ndue. Fo eaon eeal model ae been popoed o pedc e non-neonan beao of aou ype of flud. One cla of flud c a ganed condeable aenon n ecen yea e couple e flud. Couple ee ae a conequence of e aumpon a e neacon of one pa of a body on anoe, aco a uface, equalen o a foce and momen dbuon. Couple e flud con of gd, andomly oened pacle upended n a cou medum uc a blood flud, eleco-eologcal flud and ynec flud. Te man feaue of couple e flud a e e eno an-ymmec. oke [] genealzed e clacal model o nclude e effec of e peence of e couple ee and couple e model a been dely ued becaue of elae Maemacal mplcy compaed e oe model deeloped fo effec of e couple ee. T flud eoy dcued n deal by oke melf n eae Teoe of Flud Mcoucue [] een e alo peened a long l of poblem dcued by eeace efeence o eoy. Recenly, e udy of couple e flud flo a been e ubjec of gea nee, due o depead ndual and cenfc applcaon n pumpng flud, uc a ynec flud, polyme-ckened ol, lqud cyal and anmal blood. Oe mpoan feld ee couple e flud ae applcaon ae queezng and lubcaon eoy. Te moon of flud oug poou pemeable uface a lo Reynold numbe a long been an mpoan ubjec n e feld of cemcal, bomedcal, and enonmenal engneeng and cence. T penomenon fundamenal n naue and of gea paccal mpoance n many dee applcaon lke poducon of ol and ga fom geologcal ucue, e gafcaon of coal, e eong of ale ol, flaon, uface caaly of cemcal eacon, adopon, coalecence, dyng, on excange and comaogapy. ang fom Couee flo, e flo geneaed n flud by e moon of uface ae been aacng e eeace. Among em, e udy of flo due o longudnal and oonal ocllaon peen ome nee n dffeen engneeng aea lke Oceanogapy, e ecnology of baon on macney, e poce of cean polyme lqud cyal, and e offoe dllng of ol. Tee ae ee pycal uaon n c e udy of e longudnal and oonal ocllaon can be appled. Te f applcaon n lubcaon eoy. Te cylndcal beang conanng a non-neonan flud lubcan ae ubjec o longudnal and oonal baon on e macney. A econd applcaon e flo of polyme lqud cyal made of dumbbell lke molecule poceed nde a ccula cylnde c ubjec o longudnal and oonal ocllaon. And fnally, a poble d applcaon e flo of mud n e dllng of an offoe ol dllng un c ubjeced o ocllaon due o oceanc ae. Te moon of a clacal cou flud due o e oaon of an nfne cylndcal od mmeed n e flud a f decbed by oke []. Lae follong e ok, many flo poblem due o e moon of bode ee oled. ome flo poblem elaed o e moon of a cylndcal od pefomng longudnal and oonal ocllaon ae gen belo. Caaella e al., [] uded e exenal flo due o longudnal and oonal ocllaon of a od n a Neonan flud and obaned an exac oluon fo e ame. Rajagopal [5] uded e 5

2 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com ame poblem fo e cae of a econd gade flud. Ramkoon and Majumda [6] uded e nenal flo due o longudnal and oonal ocllaon of a cou flud and ey deed an analycal expeon fo eloce, ea ee and dag on e cylnde. Ramkoon e al., [7] obaned an exac oluon fo an nfne od undegong bo longudnal and oonal ocllaon n a pola flud and ey ae peened e effec of mcopola paamee on e mcooaon and elocy feld gapcally. Calmele-Eluu e al., [8] uded e nenal flo of a mcopola flud nde a ccula cylnde ubjec o longudnal and oonal ocllaon and ey ae on e effec of mcopola flud on o componen elocy feld oug gap. Oen and Raaman [9] uded e ame ype of flo an Oldoyd-B lqud. A lage numbe of eoecal negaon dealng eady ncompeble lamna flo ee njecon o ucon a e boundae ae appeaed dung e la fe decade. eeal auo, o menon ome [-] ae uded e eady lamna flo of an ncompeble cou flud n a o-dmenonal cannel paallel poou all. oundalgeka e al., [] uded e effec of couple ee on e ocllaoy flo pa poou, nfne, fla plae en e fee eam elocy ocllae n magnude abou a conan mean. Eldabe e al., [5] ae uded e effec of couple ee on an uneady MHD Eyng Poell model of non-neonan flud flo beeen o paallel fxed poou plae unde a unfom exenal magnec feld. Tey ae on e effec of couple e paamee and Hamann numbe on elocy dbuon oug gap. Deaka e al., [6] uded e oke f and econd poblem fo an ncompeble couple e flud by ung e condon a couple ee an on e bounday. Tey ae ploed e elocy pofle fo dffeen me and dffeen alue of couple e Reynold numbe. naacaya e al., [7] uded e lamna flo of a couple e flud n a poou cannel expandng o conacng all ymmec njecon o ucon along e unfomly expandng poou all by ung mlay anfomaon. Tey ae peened gap fo elocy componen and empeaue dbuon fo dffeen alue of e flud and geomec popee. Ramana Muy e al., [8] uded e eady MHD flo of a mcopola flud oug a poou ccula ppe conan ucon/njecon. Tey ae on e effec of kn fcon epec o mcopola paamee and Hamann numbe oug gap. To e exen of e knoledge of e auo, ey fe leaue ae aalable on e flo due o ocllaon of a od n couple e flud. Te poblem menoned n [7] and [8] ae ome example n decon. Hence, n pape e conde e flo of couple e flud geneaed by a poou ccula cylnde pefomng longudnal and oonal ocllaon and ubjeced o ucon elocy a e uface.. FORMULATION OF THE PROBLEM Conde a poou ccula cylnde of adu a n an nfne expanon of a couple e flud. Te cylnde ubjeced o oonal ocllaon, Exp (ω ) and longudnal ocllaon, Exp (ω ) amplude q nβ, q co β along e epece decon ee ω e fequency of e oonal ocllaon, ω e fequency of e longudnal ocllaon, q e magnude of e ocllaon and β e angle beeen e decon of oonal ocllaon and e bae eco e θ..e., e cylnde ocllae elocy a gen by e expeon ( ωτ ωτ Q q n β e e Co β e e ) Γ θ u a ucon o njecon elocy on e uface of e poou cylnde. Cylndcal pola coodnae yem condeed e Z-ax along e ax of e cylnde and ogn on e ax. Le R, θ and Z denoe e adal, azmual and axal coodnae epecely of a pon n e egon of flo. No e conde e flo geneaed n e couple e flud due o e ocllaon of e cylnde. Te pycal model lluang e poblem unde condeaon on n Fgue-. Fgue-. Geomecal epeenaon of e poblem: non-dmenonal fom. Afe neglecng body foce and body couple, e condon of ncompebly and e equaon moon fo a couple e flud, a gen by oke [] ae: Q () Q ρ Q Q P µ Q τ η Q () Wee Q elocy eco, P flud peue, z. ρ deny, τ me, µ coy and η couple e coy coeffcen, e dmenonal gaden. By 5

3 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com naue of e flo, e elocy componen ae axally ymmec and depend only on adal dance and me. Hence e elocy eco aken of e fom ( R) e V ( R,τ) eθ W ( R,τ) e z Q U () Le u noduce e follong non dmenonal ceme ωa ω σ, σ a R,, Q P q, p, q q a q ρ q q u U V τ, n, u,, W () a q q q q By e non-dmenonal ceme () n () and (), e equaon fo e flo ae anfomed o e follong non-dmenonal fom.. q (5) q Re q. q Re p q q (6) ρq ee Re a η Reynold numbe and µ µ a couple e paamee. No o mac e ocllang bounday, e elocy n () aumed n e fom σ σ () e () e eθ ( ) e ez q u (7) Te equaon (6) ll ge ae o e follong ee cala equaon n e decon of bae eco dp n d σ e (8) n Re σ D D n Re σ ee (9) and D D Ung D and D n e equaon (9), e ge a () a a ee Re n a, a, a Re σ and Ren a a mlaly e e () a () b b b () ee Ren b, b Reσ and b No e equaon () and () ae oled fo and unde e no lp condon and ype A condon o ype B condon on e bounday. Tee condon ae gen a follo.. BOUNDARY CONDITION No lp condon Te elocy of e flud on e bounday equal o e elocy of e bounday. I explcly gen by Q Velocy of Γ Γ q ω τ ω τ ( coβ e n β e e ) eθ I ake e follong n non-dmenonal fom σ q co β e e θ n β e T condon can be explcly en a n e follong equaon coβ nβ () ( ) And ( ) Type A condon Type A-condon epeen anng of couple e eno on e bounday. Te conue equaon fo couple e eno M gen by M mi η ( Q ) η [ ( Q ] T ) () Takng equaon (), e ge e expeon fo M a M m e e e e e e σ σ q e e θ e e z ( ) θ θ z z η q η q σ e e e θ a η q a a η q a e σ z e z σ e θe n e 5

4 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com η q σ η q σ D e e e z D e e ze a a If M ane on e bounday, e ge e condon a D and e ee e η η on (5) Type B condon Type B-condon e upe adeence condon on e bounday. T condon eque a angula elocy of e flud pacle on e bounday equal o angula elocy of e bounday..e., ω Γ QΓ mple a d σ And ( ) d (6) A, e flud a e and bounday condon can be aken a D e d d ( ) on (7). FINITE DIFFERENCE METHOD OF OLUTION In e of e complcaed naue of o equaon () and (), e analycal oluon fo and eem o be beyond eac. Te deal of fne dffeence meod ued ee can be uded fom Ref. [9], fo obanng e oluon fo and. We ake 5 un of dance fom ogn ey lage epeenng nfny. Hence e dcee e neal [, 5] no n ubneal n node. Eac node epeened by, 9/n e ep leng, ang fom f node o e la node n 5. Te alue of e funcon, a ae gen by and. Te ymmec deae fomulae a e node ae gen a belo: 6 (8) ubung ee deae gen n (8), n e equaon () e ge,,,, 5, (9) ee, ( ), a ( ) ( ) a, 6 a ( a ) a ( ), a ( ) a () 5, Te fne dffeence fom of () a e follong:,,,, 5, (), ( ), b ( ) ( b ) b b, 6 ( ), b ( b ) () 5, Type A oluon fo eloce and We ake e follong bounday condon ( ) coβ n D, n, and D () Ealuang (9) fo dffeen alue of e oban : : :,,, 5,,,,, 5,, 5

5 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com,,, 5,, :,,,, 5, 5 n :, n n,n n,n n 5,n n, n n n:, n n,n n,n n 5,n n,n n () Tu e yem of equaon () epeen n equaon n n unknon. Hence e eque o moe equaon. Tee can be obaned fom D And D (5) Ung e bounday condon (5), en e ae and n 5 n 6 n (6) Wee,,, n n n, 5 n and 6 n Expeng e equaon () and (6) n max fom a A X B (7) ee e mace A, X and B ae gen n e appendx. B conan and n c ae e alue of on e bounday and X con of alue of n e egon. olng e yem (7) e ge e oluon fo. No e fnd oluon fo by applyng e bounday condon () nβ n n, n, e and e (8) Ealuang () fo dffeen alue of e oban :,,, 5,, :,,, 5,, :,,, 5,, :,,,, 5, 5, n n,n n,n n 5,n n, n n n:, n n,n n,n n 5,n n, n n (9) Tu e yem of equaon (9) epeen n equaon n n unknon. Hence e eque o moe equaon c ae obaned fom e bounday condon e en e ae and e, and n n n n () e e Wee,, e e n and n Expeng e equaon (9)-() n max fom a A X B () ee e mace A, X and B ae gen n e appendx. B conan and n c ae e alue of on e bounday and X con of alue of n e egon. olng e yem () e ge e oluon fo Type B oluon fo eloce and We ake e follong bounday condon ( ) coβ n, n, ( ) d d ( ) d d d and d Ung e bounday condon ( ) d and ( ), e ge d And n n n () Fom (9) ubung,,..n, n and fom (), ng n max e ge a A X B () ee A, B ae gen n appendx. n n : 55

6 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com mlaly e ue e follong bounday condon fo elocy a () nb n n, n, σ and Ung e condon σ and, e ge σ and () n n Fom () ubung,,..n, n and fom (), ng n max e ge a A X B (5) ee A and B ae gen n e appendx. On olng (5), e ge e oluon fo X, e alue of axal elocy. 5. DRAG ON THE CYLINDER Te dag D acng on a cylnde of leng L gen by D L π a ( T co β T n β ) d θ (6) Te e componen n (6) and Couple e eno M ae defned by e follong conue equaon fo couple e flud (oke []). T j PI λ ( Q ) I T µ ( Q ( Q ) ) I ( M ) ( Q ) η [ ( Q )] (7) T M mi η (8) Te e componen T and T on e cylnde can be calculaed a µq T e a (9) µq σ T e a () Applyng e fne dffeence ceme fo (9) and (), e non-dmenonal fom of e componen ae calculaed a σ T T µq a e σ σ µq e a Te non-dmenonal dag can be calculaed fom (6) a T n D () Wee co β T β D D Lπµ q 6. REULT AND DICUION Te analycal expeon fo e nondmenonal elocy componen, and dag ae gen by e equaon (), () and () epecely. Tee alue depend on e alue of β, f β, e ge only oonal ocllaon and f β π/, e ge only axal (Longudnal) ocllaon. Te numecal eul ae peened n e fom of gap fo, Re., σ.5, σ.5, β.7, n.6, π. Te Fgue fo ype A bounday condon ae on on lef column and e Fgue fo ype B condon ae on e g column. Te eloce and a dffeen Reynold numbe ype A and ype B bounday condon ae on n Fgue -5. We noce a a Reynold numbe Re nceae bo e eloce and deceae. Te eloce and a dffeen nondmenonal me ae on n Fgue 6-9. We obee a e anee and axal elocy componen nea e cylnde ae deelopng and flucuang aound zeo e ame fequency a e cylnde. A e a of a cycle, e flo a maxmum eloce locaed a e uface of cylnde, a gadual deceae oad zeo n e egon aay fom e cylnde. A e cycle connue, e eloce deceae e maxmum alue no longe a e cylnde uface bu nde e flo feld. Fom Fgue -, e ee a a e couple e paamee nceae, e anee elocy deceae fo ype A condon and nceae fo ype B condon and e axal elocy nceae fo ype A condon, le deceae fo ype B condon. Type B bounday condon doen nole e paamee e, c e ao of couple e coy coeffcen η and η'. In 56

7 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com ype A condon, a can be een fom Fgue -5 bo e axal and anee eloce ae ngnfcan e. Te non-dmenonal dag calculaed numecally fo dffeen alue of non-dmenonal me n mulple of π/σ a fxed alue of σ, σ and e eul ae on n e Fgue 6-7. In e equaon of moon, local acceleaon em domnae f σ, σ ae lage. To ae all em of LH n e ame ode n equaon (9) and (), e fequency paamee σ, σ ae o be mall. Hence e ake bo σ, σ <. In e calculaon of dag alo e obee a f σ <, σ <, e dag ll be n eaonable alue. Fom Fgue 6-7, can be een a a e couple e paamee nceae, e amplude of ocllaon fo dag nceae fo bo e ype A and ype B condon. I can be een fo bo e ype A and ype B condon a σ nceae, e aaon n dag a e cylnde all ae cangng n amplude and fequency (Fgue 8-9). Fom Fgue- and Fgue-, obeed a e dag ngnfcan o e aaon n σ fo ype a condon and fo ype B condon e dag ocllae egulaly a σ nceae. Fom Fgue -, e noe a a Reynold numbe Re nceae magnude of dag nceae fo mall alue of ucon; bu fo ge alue of ucon dag nceae fo mall alue of Reynold numbe and en deceae a ge alue and almo conan fo ey g alue of ucon ae. Fom Fgue-, e noe a dag ngnfcan o e aaon n e, fo ype a condon and ype B condon ndependen of e. Fgue-. Type B aaon of. Fgue-. Type A aaon of. 7. CONCLUION We ae obeed a:. Te flo ene epec o couple e paamee and ype A and ype B condon o oppoe end..e., e anee elocy deceae fo ype A condon, nceae fo ype B condon;. Te dag nceae a nceae.e. e dag offeed by cou flud le an a of couple e flud; and. ucon on e cylnde deceae e dag. Fgue-5. Type B aaon of. Fgue-. Type A aaon of. 57

8 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com Fgue-6. Type A aaon of Exp(σ ). Fgue-9. Type B aaon of Exp (σ ) Fgue-. Type A aaon of. Fgue-7. Type B aaon of Exp (σ ). Fgue-. Type B aaon of. Fgue-8. Type A aaon of Exp (σ ). 58

9 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com Fgue-. Type A aaon of. Fgue-5. Type A aaon of. Fgue-. Type B aaon of. Fgue-6. Type A aaon of D' σ. Fgue-. Type A aaon of. Fgue-7. Type B aaon of D' σ. 59

10 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com Fgue-8. Type A aaon of D'. Fgue-. Type B aaon of D'. Fgue-9. Type B aaon of D'. Fgue-. Type A aaon of D' Re. Fgue-. Type A aaon of D'. Fgue-. Type B aaon of D' Re. 6

11 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com [] A.. Beman. 95. Lamna flo n cannel poou all. J. App. Py. : -5. [] J.R. ella Lamna flo n cannel poou all a g ucon Reynold numbe. J. Appl. Py. 6: [].W. Yuan Fue negaon of lamna flo n cannel poou all. J. Appl. Py. 7: REFERENCE Fgue-. Type A aaon of D'. [] V.K. oke Couple e n flud. Py. Flud. 9: [] V.K. oke. 98. Teoe of Flud Mcoucue. pnge, Ne Yok. [] G.G. oke On e effec of oaon of cylnde and pee abou e on axe n nceang e logamc decemen of e ac of baon. Cambdge Uney Pe, England. pp. 7-. [] M.J. Caaella and P.A. Laua Dag on ocllang od longudnal and oonal Moon. J. Hydonau. : 8-8. [5] K.R. Rajagopal. 98. Longudnal and Toonal ocllaon of a od n a non-neonan flud. Aca. Mec. 9: [6] H. Ramkoon and.r.majumda. 99. Flo due o e longudnal and onal ocllaon of a cylnde. ZAMP. : [7] H. Ramkoon, C.V. Eaaan and.r. Majumda. 99. Longudnal and Toonal ocllaon of a od n a pola flud. In. J. Engng. c. 9(): 5-. [8] C. Calmele-Eluu and D.R. Mazumda Flo of Mcopola flud oug a ccula cylnde ubjec o longudnal and onal ocllaon. Mal. Compu. Modelng. 7: [9] D. Oen and K. Raaman. 6. On e flo of an Oldoyd-B lqud oug a ag ccula ube pefomng longudnal and oonal ocllaon of dffeen fequence. Maemaca. : -9. [] R.M. Tel, G.M. ea Lamna flo oug paallel and unfomly poou all of dffeen pemeably. ZAMP. 6: 7-8. [] V. M. oundalgeka and R. N. Aanake. 97. Effec of couple ee on e ocllaoy flo pa an nfne plae conan ucon. Meccanca. pp [5] N. T. M. Eldabe, A. A. Haan and Mona A. A. Moamed.. Effec of Couple ee on e MHD of a Non-Neonan Uneady Flo beeen To Paallel Poou Plae. Z. Naufoc. 58a. -. [6] M. Deaka, T.K.V. Iyenga. 8. oke Poblem fo an Incompeble Couple e Flud. Nonlnea Analy: Modelng and Conol. (): 8-9. [7] D. naacaya, N. naacayulu, O. Odelu. 9. Flo and ea anfe of couple e flud n a poou cannel expandng and conacng all. Inenaonal Communcaon n Hea and Ma Tanfe. 6: [8] J.V. Ramana Muy and N.K. Baal. 9. eady flo of mcopola flud oug a ccula ppe unde a anee magnec feld conan ucon/njecon. In. J. of Appl. Ma and Mec. 5(): -. [9] P. Muu, B.V. Ra Kuma and Peeyu Canda. 8. Pealc moon of mcopola flud n cylndcal ube: effec of all popee. Appl. Ma. Modelng. : 9-. Appendx Te mace A, X, B ; A, X, B ; A, B and A, B defned eale n e equaon (7), (), () and (5) ae gen by e follong expeon. Te coeffcen max A gen n (7) defned a: 6

12 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com, A,,,,,, 5,,,, 5,,, 5,, 5,,n,n,n,n,n,n,n,n,n,n 5,n,n,n,n 5,n,n 5 5,n Te oluon eco X fo elocy a dffeen pon gen by X [,,,,,. n 5, n, n, n, n, n, n, n ] T and e eco B fo e bounday alue gen by B [,,,,,,,..., 5, n n,, n n,, n n, 6 n ] T Te mace A, X, B defned n equaon () ae a follo. Te coeffcen max A gen by, A,,,,,, 5,,,, 5,,, 5,, 5,,n,n,n,n,n,n,n,n,n,n 5,n,n,n,n 5,n,n 5,n Te oluon eco X fo elocy a dffeen pon gen by X [,,,,,, 5.. n 5, n, n, n, n, n, n, n ] T and e eco B fo e bounday alue gen by B [,,,,,,,, 5, n n,, n n,, n n, n n ] T Te coeffcen max A defned n equaon () gen by 6

13 VOL. 5, NO. 5, MAY IN ARPN Jounal of Engneeng and Appled cence 6- Aan Reeac Publng Neok (ARPN). All g eeed..apnjounal.com, A,,,,,, 5,,,, 5,,, 5,, 5,,n,n,n,n,n,n,n,n,n,n 5,n,n,n,n 5,n,n 5,n and e eco B fo e bounday alue gen by B [(/ ),,,,,,,,.,, 5,n n,,n n,,n n, (/ n ) n ] T Te coeffcen max A defned n equaon (5) gen by, A,,,,,, 5,,,, 5,,, 5,, 5,,n,n,n,n,n,n,n,n,n,n 5,n,n,n,n 5,n,n 5,n and e eco B fo e bounday alue gen by B [σ,,,,,,,,.,, 5,n n,,n n,,n n,] T 6